After Wiles' proof of the Shimura-Taniyama conjecture, the Birch and
Swinnerton-Dyer conjecture has become the outstanding open problem in
the arithmetic theory of elliptic curves. In the late 80's, the work
of
Kolyvagin led to an almost complete proof of the Birch and
Swinnerton-Dyer conjecture for elliptic curves of (analytic) rank at
most one. The higher rank case remains shrouded in mystery. Recently
Bertolini and I have been able to prove part of a p-adic variant of
the
Birch and Swinnerton Dyer conjecture which applies to curves of higher
rank. The proof combines ideas that arose in the work of Wiles and
earlier work of Ribet on Fermat's Last Theorem with the methods of
Thaine and Kolyvagin.